A363499 -id:A363499 - cp's OEIS frontend

A174060 a(n) = Sum_{k=0..n} floor(sqrt(k))^2.

0, 1, 2, 3, 7, 11, 15, 19, 23, 32, 41, 50, 59, 68, 77, 86, 102, 118, 134, 150, 166, 182, 198, 214, 230, 255, 280, 305, 330, 355, 380, 405, 430, 455, 480, 505, 541, 577, 613, 649, 685, 721, 757, 793, 829, 865, 901, 937, 973, 1022, 1071, 1120, 1169, 1218, 1267, 1316

Offset: 0

Vladimir Joseph Stephan Orlovsky, Mar 06 2010

Partial sums of A048760. - R. J. Mathar, Mar 31 2010

Karl-Heinz Hofmann, Table of n, a(n) for n = 0..10000

Cf. A022554 (1st), this sequence (2nd), A363497 (3rd).

Cf. A363498 (4th), A363499 (5th), A048760.

Accumulate[Table[Floor[Sqrt[k]]^2, {k, 0, 59}]] (* Harvey P. Dale, Jul 13 2013 *)

a(n)=my(m=sqrtint(n+1));(n+1)*m^2-m*(m+1)*(3*m^2+m-1)/6 \\ Charles R Greathouse IV, Jul 04 2013

a(n) = sum(k=0, n, sqrtint(k)^2); \\ Karl-Heinz Hofmann, Jun 15 2023

from math import isqrt
A174060 = [0]
for n in range(1,56): A174060.append(A174060[-1] + isqrt(n)**2)
print(A174060) # Karl-Heinz Hofmann, Jun 15 2023

from math import isqrt
def A174060(n): return ((m:=isqrt(n+1))*(6*m*(n+1) - (m+1)*(3*m**2+m-1)))//6
# Karl-Heinz Hofmann, Jun 15 2023

a(n) = (1/6)*m*(6*m*n - (m+1)*(3*m^2+m-1)) with m = floor(sqrt(n)). - Yalcin Aktar, Jan 30 2012

A363497 a(n) = Sum_{k=0..n} floor(sqrt(k))^3.

Original entry on oeis.org

0, 1, 2, 3, 11, 19, 27, 35, 43, 70, 97, 124, 151, 178, 205, 232, 296, 360, 424, 488, 552, 616, 680, 744, 808, 933, 1058, 1183, 1308, 1433, 1558, 1683, 1808, 1933, 2058, 2183, 2399, 2615, 2831, 3047, 3263, 3479, 3695, 3911, 4127, 4343, 4559, 4775, 4991, 5334

Offset: 0

Hans J. H. Tuenter, Jun 05 2023

Partial sums of the third powers of the terms of A000196.

Karl-Heinz Hofmann, Table of n, a(n) for n = 0..10000

Sums of powers of A000196: A022554 (1st), A174060 (2nd), this sequence (3rd), A363498 (4th), A363499 (5th).

Table[(n + 1) #^3 - (1/60) # (# + 1) (3 # - 1) (12 #^2 + 7 # - 4) &[Floor@ Sqrt[n]], {n, 0, 50}] (* Michael De Vlieger, Jun 10 2023 *)

a(n) = sum(k=0, n, sqrtint(k)^3); \\ Michel Marcus, Jun 06 2023

from math import isqrt
A363497 = [0]
for n in range(1,50): A363497.append(A363497[-1] + isqrt(n)**3)
print(A363497) # Karl-Heinz Hofmann, Jun 14 2023

from math import isqrt
def A363497(n):return (m:=isqrt(n))**3*(n+1)-(m*(m+1)*(3*m-1)*(12*m**2+7*m-4))//60
# Karl-Heinz Hofmann, Jun 14 2023

a(n) = (n+1)*m^3 - (1/60)*m*(m+1)*(3*m-1)*(12*m^2+7*m-4), where m = floor(sqrt(n)).

A363498 a(n) = Sum_{k=0..n} floor(sqrt(k))^4.

Original entry on oeis.org

0, 1, 2, 3, 19, 35, 51, 67, 83, 164, 245, 326, 407, 488, 569, 650, 906, 1162, 1418, 1674, 1930, 2186, 2442, 2698, 2954, 3579, 4204, 4829, 5454, 6079, 6704, 7329, 7954, 8579, 9204, 9829, 11125, 12421, 13717, 15013, 16309, 17605, 18901, 20197, 21493, 22789

Offset: 0

Hans J. H. Tuenter, Jun 05 2023

Partial sums of the fourth powers of the terms of A000196.

Karl-Heinz Hofmann, Table of n, a(n) for n = 0..10000

Sums of powers of A000196: A022554 (1st), A174060 (2nd), A363497 (3rd), this sequence (4th), A363499 (5th).

Table[(n + 1) #^4 - (1/30) # (# + 1)*(20 #^4 + 4 #^3 - 14 #^2 + 4 # + 1) &[Floor@ Sqrt[n]], {n, 0, 45}] (* Michael De Vlieger, Jun 10 2023 *)

from math import isqrt
def A363498(n):
    return (m:=isqrt(n))**4 *(n+1) - (m*(m+1)*(20*m**4+4*m**3-14*m**2+4*m+1))//30
print([A363498(n) for n in range(0,46)]) # Karl-Heinz Hofmann, Jul 15 2023

a(n) = (n+1)*m^4 - (1/30)*m*(m+1)*(20*m^4+4*m^3-14*m^2+4*m+1), where m = floor(sqrt(n)).

cp's OEIS Frontend

A174060 a(n) = Sum_{k=0..n} floor(sqrt(k))^2.

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A363497 a(n) = Sum_{k=0..n} floor(sqrt(k))^3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Python

Formula

A363498 a(n) = Sum_{k=0..n} floor(sqrt(k))^4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Formula